Question

A certain drug is effective for an average patient only if there is at least $1\mathrm{mg}$ per $\mathrm{kg}$ in the patient's system, while it is safe only if there is at most $2\mathrm{mg}$ per $\mathrm{kg}$ in an average patient's system. Suppose that the amount in a patient's system diminishes by a multiplicative factor of $0.9$ each hour after a dose is administered. Find the maximum interval $N$ of hours between doses, and corresponding dose range $d$ (in $\mathrm{mg}/\mathrm{kg}$ ) for this $N$ that will enable use of the drug to be both safe and effective in the long term.

Short answer

6 hours, dose between 1.88 mg/kg and 2 mg/kg.

Step-by-step solution

Understand the Problem

We need to ensure the drug remains effective ($\ge 1\text{ mg/kg}$) and safe ($\le 2\text{ mg/kg}$) over a span of time given it decreases by a factor of 0.9 each hour. We are looking for the maximum interval of hours between doses ($N$) and the corresponding dose range that satisfies these conditions.

Setup the Mathematical Model

We model the concentration of the drug in the system over time after a dose. Let $d$ be the amount of drug given per dose in mg/kg. After 1 hour, the drug concentration will be $0.9d$, after 2 hours $(0.9{)}^{2}d$, and generally after $N$ hours $(0.9{)}^{N}d$. We need this concentration to be at least $1\text{ mg/kg}$ to be effective at the end of the dosing interval and no more than $2\text{ mg/kg}$ right after dosing.

Find Inequalities for Effectiveness and Safety

For effectiveness right before the next dose:$$(0.9{)}^{N}d\ge 1$$For safety right after the dose:$$d\le 2$$

Solve for Maximum Interval $N$

Substitute the safety condition into the effectiveness condition:$$(0.9{)}^N\cdot 2\ge 1$$$$(0.9{)}^N\ge 0.5$$$$N\log (0.9)\ge \log (0.5)$$$$N\le \frac{\log (0.5)}{\log (0.9)}\approx 6.58$$Since $N$ must be an integer, the maximum $N$ is 6.

Determine the Corresponding Dose Range $d$

With $N=6$, substitute back into the effectiveness condition:$$(0.9{)}^{6}d\ge 1$$$$0.531441d\ge 1$$$$d\ge \frac{1}{0.531441}\approx 1.88$$Thus, the dose range satisfying both conditions is:$$1.88\le d\le 2\text{ mg/kg}$$

Conclusion

The maximum interval $N$ between doses is 6 hours, and the corresponding dose range $d$ is between 1.88 mg/kg and 2 mg/kg.

Key concepts

Exponential Decay

Exponential decay is a process where a quantity reduces at a consistent rate over time. It is characterized by a multiplicative factor that is less than one. In the context of drug dosing, this factor describes how quickly the concentration of the drug decreases in the body.

Imagine you have a certain concentration of a drug in the body. Each hour, the effectiveness of the drug decays by a specific percentage. For instance, if the decay factor is 0.9, every hour, only 90% of the previous amount remains. Consequently, after one hour, if you had a dose of 10 mg/kg, only 9 mg/kg would be effective within the body.

This type of decay can be expressed using an exponential function, where after each hour, the remaining concentration is calculated by multiplying the initial dose by the decay factor raised to the power of the number of hours. Therefore, the concentration formula is:

• After 1 hour: $0.9d$
• After 2 hours $(0.9{)}^{2}d$
• After $N$ hours: $(0.9{)}^{N}d$

Understanding this decay pattern helps in determining how often a drug should be administered to remain effective.

Inequalities

Inequalities are mathematical expressions used to show that one value is larger or smaller than another. They are crucial in drug dosing calculations to ensure both safety and efficacy.

In drug calculations, inequalities help to establish the range within which a drug dose remains effective but does not become toxic. For a drug to be effective, its concentration in the system should be greater than or equal to a certain threshold. Concurrently, to prevent toxicity, the concentration should not exceed a higher maximum value.

In this situation, two inequalities are particularly significant:

• For effectiveness: $(0.9{)}^{N}d\ge 1$
• For safety: $d\le 2$

These inequalities define a safe and effective dosing range. Solving them requires analyzing both the rate of decay of the drug's effectiveness and the safety boundaries. When both conditions are met, the drug remains within the therapeutic range.

Pharmacokinetics

Pharmacokinetics involves the study of how drugs are absorbed, distributed, metabolized, and excreted by the body. It combines various principles and scientific methods to understand the effects and lifespan of a drug within the human body.

An important part of pharmacokinetics is determining the right dose and timing to maintain the drug's concentration within the therapeutic window. This is where the decay rate of the drug (exponential decay) and concentration limits (inequalities) come together.

The therapeutic window is the concentration range between the minimum effective concentration (MEC) and the minimum toxic concentration (MTC). Drugs should be administered such that their concentration remains within this range as long as possible:

• MEC for effectiveness is $\ge 1\text{ mg/kg}$
• MTC for safety is $\le 2\text{ mg/kg}$

In this problem, the decay in drug concentration and the prescribed inequalities ensure the drug's presence in the therapeutic window over time. Through understanding pharmacokinetics, clinicians can tailor drug schedules to optimize therapeutic effects while minimizing risks.